59] CALCULATING THE TRAJECTORIES OF SHOT. 479 



and therefore 



= tan y + ~ (a - )' (sec y) 2 + (sec y) 2 tan y ; 



in which the term involving f"(y) has disappeared. 

 Now, since /(</>) = u l (sec </>) m , we have, as before 



and therefore 



f'M 



J j.-^' = I ( : ~ ) + m tan y. 

 Hence 



/3 



and in the particular case where 1 = 2, and m = 2, we have 



tan yl ; 



J 



= tan {y + ^ (a - ' [2 (^ + 3 tan y]} . 

 Hence the angle which the chord of the arc makes with the axis of x is 



i , r / du \ i 



y + - (a - /BY [2 ^ d ^ + 3 tan yj = y, suppose. 

 Multiplying by the value of X found above, we have 



+tan y W 1 n + 4 (sec y) 2 - 6 (sec y) 2 - n - 1 



or 



